Let's solve the given problems step by step using the provided functions:

We have:

• $f(x) = 2x^2$
• $g(x) = 3x + 1$

1. $f(x) + g(x)$

$f(x) + g(x) = 2x^2 + (3x + 1) = 2x^2 + 3x + 1$

2. $f(x) - g(x)$

$f(x) - g(x) = 2x^2 - (3x + 1) = 2x^2 - 3x - 1$

3. $f(g(x))$

Substitute $g(x) = 3x + 1$ into $f(x) = 2x^2$: $f(g(x)) = 2(3x + 1)^2 = 2[(3x + 1)(3x + 1)]$ Expanding the square: $(3x + 1)(3x + 1) = 9x^2 + 6x + 1$ So: $f(g(x)) = 2(9x^2 + 6x + 1) = 18x^2 + 12x + 2$

4. $f(x)^2$

$f(x)^2 = (2x^2)^2 = 4x^4$

5. $f(x)^3 - g(x) + g(x)^2$

First, calculate $f(x)^3$, $g(x)$, and $g(x)^2$:

• $f(x)^3 = (2x^2)^3 = 8x^6$
• $g(x) = 3x + 1$
• $g(x)^2 = (3x + 1)^2 = 9x^2 + 6x + 1$

Now, combine them: $f(x)^3 - g(x) + g(x)^2 = 8x^6 - (3x + 1) + (9x^2 + 6x + 1)$ Simplifying: $= 8x^6 - 3x - 1 + 9x^2 + 6x + 1 = 8x^6 + 9x^2 + 3x$

Let me know if you want further clarification or details!

Relative Questions:

1. What is the general process for finding $f(g(x))$ for different types of functions?
2. How would the result of $f(x)^2$ change if $f(x)$ was a linear function instead of quadratic?
3. Can you describe a scenario where the combination $f(x) - g(x)$ represents a real-world problem?
4. How do we expand binomials like $(3x + 1)^2$ efficiently?
5. What are the general rules for combining functions like $f(x)$ and $g(x)$?

Tip:

Always simplify functions step by step, especially when combining terms or applying operations like squaring or cubing.